A corrector theory for diffusion-homogenization limits of linear transport equations∗ Guillaume Bal † Naoufel Ben Abdallah Marjolaine Puel ‡ 2 1 January 24, 2012 0 2 n a J Abstract 1 2 Thispaperconcerns thediffusion-homogenization of transportequations when ] both the adimensionalized scale of the heterogeneities α and the adimensionalized P mean-free path ε converge to 0. When α = ε, it is well known that the heteroge- A neous transport solution converges to a homogenized diffusion solution. We are . h interested here in the situation where 0 < ε ≪ α ≪ 1 and in the respective rates t a of convergences to the homogenized limit and to the diffusive limit. Our main m result is an approximation to the transport solution with an error term that is [ negligible compared to the maximum of α and ε. After establishing the diffusion- α 1 homogenization limit to the transport solution, we show that the corrector is v dominated by an error to homogenization when α2 ≪ ε and by an an error to 4 diffusion when ε ≪ α2. 2 4 Our regime of interest involves singular perturbations in the small parameter 4 η = ε. Disconnected local equilibria at η = 0 need to be reconnected to provide . α 1 a global equilibrium on the cell of periodicity when η > 0. This reconnection 0 between local and global equilibria is shown to hold when sufficient no-drift con- 2 1 ditions are satisfied. The Hilbert expansion methodology followed in this paper : builds on corrector theories for the result developed in [9]. v i X r 1 Setting of the problem a This paper studies the interaction between the convergence to a homogenized limit and the convergence to a diffusive limit in the context of linear transport equations, or linear Boltzmann equations [12, 6], that model the propagation of particles in scattering environments. These two phenomena have been widely studied in the past [11, 1, 16, 23, 21, 22, 24, 14]. The homogenization limit typically arises when the underlying ∗NBA and MP were partially supported by ANR project BLAN07-2 212988. GB was partially supportedby NSFGrantDMS-0804696. GB wouldlike to thankthe Universit´ePaulSabatierfor their hospitality in the Spring of 2009, when part of this researcheffort was conducted. †Department of Applied Physics and Applied Mathematics, Columbia University, New York NY, 10027; [email protected] ‡Institut de Math´ematiques, Universit´e de Toulouse and CNRS, Universit´e Paul Sabatier, 31062 Toulouse Cedex 9, France; [email protected]. 1 coefficients oscillate at a scale α ≪ 1 much smaller than the macroscopic scale at which phenomena are observed. The diffusion limit arises in highly scattering environments in which an equilibrium emerges in the velocity variable and a diffusion equation models the spatial behavior of the probability density of particles. High scattering is modeled by a mean free path ε ≪ 1, where ε is the (adimensionalized) main distance between successive collisions with the underlying medium. We assume here that the oscillations in the medium are periodic with period α ≪ 1. When both α and ε converge to 0, we expect the transport solution to converge to the solution ofahomogenized diffusion equation. How fastconvergence mayoccur andwhat are the main contributions to the error between the heterogeneous transport solution and its homogenized limit are the main problems of interest in this paper. We restrict ourselves to the case where the mean free path is (much) smaller than the correlation length α. We thus set η = ε and we assume that η ≪ 1. The equation for the particle α density takes the form: 1 x v ·∇fε +εfε + Q fε = εS (x,v) := εS(x, ,v), (1) α α ε α α α α in an infinite domain Rd × V where V is a smooth, symetric, compact domain in the velocity space such that 0 6∈ V. This equation could also be seen as an evolution equation by a change of variables uε = e−tfε in which the source S would play the role α α of an initial condition. We restrict ourselves to the time independent setting and to the case of constant absorption to simplify. The source term S is 1−periodic in y = x. We define the collision operator as α x x Q f = Σ( ,v)f(x,v)− σ( ,v′,v)f(x,v′)dν(v′). (2) α α Z α V The choice of Σ is such that the above operator is conservative in the following sense. Define Qf(y,v) = Σ(y,v)f(y,v)− σ(y,v′,v)f(y,v′)dν(v′), y ∈ Y, (3) Z V where Y = [0,1]d is the unit cell. Then we assume the existence of C−1 ≥ ψη(y,v) ≥ C > 0 such that (ηv·∇ +Q)ψη = 0, Y ×V. (4) y We also define (−ηv ·∇ +Q∗)ψη∗ = 0, Y ×V, (5) y where Q∗ is the adjoint operator to Q defined for a.e. y ∈ Y and g ∈ L2(V) as Q∗g(v) = Σ(y,v)g(v)− σ(y,v,v′)g(v′)dν(v′). (6) Z V The simultaneous limit when the mean free path and the correlation length go to zero together with α = ε has been considered in several recent papers [2, 3, 4, 5, 10, 17, 18, 19, 20, 25]. In all those paper, only the case η = 1 is considered except in [25], where both ψη and ψη∗ are space independent and hence also independent of η. A first result in the case η ≪ 1 was obtained in [9] for spatially dependent ψη in the setting where 2 ψ∗,η ≡ 1. In that paper, a two scale convergence result is established in the setting where the limiting behavior involves the homogenization of a heterogeneous diffusion. The proof of that result is based on the weak formulation of the transport equation and on the method of moments. The aim of the present work is to implement the Hilbert expansion method, which leads to a strong convergence result under appropriate smoothness conditions in the (restricted) setting where ψ∗,η = ψ∗(v). This strong convergence has two advantages. First, we are able to compute the correction terms in the expansion of the transport solution up to a term that is negligible compared to the maximum of α and η = ε. α Second, we expect this result also to be the first step in the study of the nonlinear Boltzmann equation given by the Fermi-Dirac equation in the spirit of [7, 8]. Note that inthe casewhere η goestoinfinity, we expect the limit to bethediffusion approximation to the homogenized transport equation obtained in [15]. The main mathematical difficulty of the present work arises because the limit η → 0 in (4) is singular. In the limit η = 0, local equilibria are obtained as a function of v ∈ V for each position y ∈ Y. For η > 0, a global equilibrium on Y ×V emerges, as in the standard procedure obtained when η = 1 [2, 19, 17, 21, 22, 25]. The passage from the local equilibria to the global equilibrium may in fact be arbitrarily complicated. Several conditions need to be imposed in order for a well-defined macroscopic equilibrium to arise. Even in the case η = 1 do we need to impose a no-drift condition. In the presence of drift, advection dominates scattering and entirely different phenomena arise (see, e.g., [17], [5] and the derivation of Euler equations when advection is dominant). Under appropriatesufficient symmetry assumptionssimilar to(thoughmoreconstrainingthan) those in[2], we areableto verify the necessary no-drift conditions used inour derivation. The rest of the paper is structured as follows. The main hypotheses of regularity and no-drift as well as the main results of this paper are described in section 2. The main result is Theorem 2.1 below. The rest of the paper is devoted to its proof. Global a priori estimates are formulated and proved in section 3. The expansion of the transport solution in powers of ε is treated in section 4. The expansion in η of several cell solutions is given in section 5. The definition and η-dependence of spatial density terms are given in section 6. These results are combined to finish the proof of the theorem in section 7. Several technical results obtainedin[9] andthegeneralizationoftheir proofsifnecessary are collected in the Appendix. 2 Main result Under regularity assumptions recalled below, a standard application of a Banach fixed point theorem [13] ensures that (1) admits a unique solution in Hk(Rd,L2(V)). The limit of fε,η(x,v) := fε(x,v) as η → 0, however, involves singular perturbations. The α reason is that the local equilibria in (4) and (5) become degenerate in the limit η → 0. In this limit, equilibria at different points y ∈ Y become disconnected and this can result in a very large effect at the macroscopic scale x. We consider here situations where the equilibria remain smooth in the y variable and generate no drift. Drift effects are ubiquitous in the homogenization of transport equations, with drastic effects as may be seen in, e.g., [5]. Diffusion limits arise under sufficient no-drift conditions as in, e.g., [2], which ensure that transport is not in an advection-dominated regime. Our 3 analysis in this paper shows that diffusion-like regimes are still valid in the singular limit η → 0 under appropriate no-drift assumptions. We show that these assumptions are consequences of symmetries of the transport coefficients, which we now define. Ourfirst mainassumptions isthat ψ∗,η(y,v)isindependent ofy andhence of η. Note that Q in (4) is independent of η and it is therefore not clear why non-trivial solutions would exist for all values of η. Here, we assume that a.e. y ∈ Y, we have an equilibrium ψ∗(v) solution of Q∗(ψ∗) = 0 in V. (7) The equilibrium solution is allowed to depend on v but is independent of y ∈ Y. When σ(y,v′,v) and Σ(y,v) are continuous functions bounded above and below by positive constants, then ψ∗ can bechosen positive andnormalized so that (ψ∗)2(y,v)dν(v) = 1 V a.e. y ∈ Y. Moreover, up to normalization, ψ∗(v) is the uniqueR solution to (7) as an application of the Krein Rutman theorem [2, 13] for the compact operators defined for a.e. y ∈ Y: 1 f 7→ σ(y,v,v′)f(v′)dν(v′), (8) Σ(y,v) Z V which preserve the (solid) cone of positive continuous functions. The no-drift conditions mentioned above will be verified under sufficient symmetry assumptions. We first assume that: σ(y,v′,v) = σ(y,−v′,−v) = σ(−y,v′,v), Σ(y,v) = Σ(y,−v) = Σ(−y,v). (9) We deduce that ψ∗(−v) = ψ∗(v). Note that a method to construct local equilibria consists of selecting σ satisfying the above symmetries, ψ∗ arbitrary (uniformly positive) such that ψ∗(−v) = ψ∗(v), and finally define Σ(y,v) by (7), which also satisfies (9). We also obtain the existence of a unique, bounded, positive, solution ψ(y,v) of the adjoint equation Q(ψ(y,·)) = 0, in V, (10) normalized so that ψ(y,v)ψ∗(y,v)dν(v) = 1 a.e. y ∈ Y. It is not difficult to observe V that ψ(y,−v) = ψ(yR,v) = ψ(−y,v) when (9) holds. The Krein Rutman theorem for (8) (all eigenvalues not equal to 1 have modulus strictly smaller than 1) shows that 0 is a simple eigenvalue associated to Q and Q∗ and that all other eigenvalues of Q and Q∗ have strictly positive real part [2, 13]. On the vector space of functions f ∈ L2(V) such that f(v)ψ∗(v)dν(v) = 0, we define V ∞R Q−1f := − e−rQfdr, (11) Z 0 which converges a.e. y ∈ Y thanks to the spectral gap we just mentioned. Note that (Q−1f,ψ∗) = 0 by construction. The inverse operator Q−∗ := (Q∗)−1 is defined L2(V) similarly. We verify that under (9), both Q−1 and Q−∗ preserve the subspaces of even and odd functions in the variable v. With ψ∗ seen as a normalized solution of (−ηv ·∇ +Q∗)ψ∗ = 0 on Y ×V, we also y obtain the existence of unique, bounded, positive, solutions ψη(y,v) of (4) normalized such that ψη(y,v)ψ∗(v)dν(v) = 1. Upon introducing Tη = ηv ·∇ + Q, we also Y×V y define the Rinverse operator ∞ Tη−1f = − e−rTηfdr, f ∈ L2(Y ×V) s.t. f(y,v)ψ∗(v)dydν(v) = 0. (12) Z Z 0 Y×V 4 Finally, we observe that under (9), ψη(y,−v) and ψη(−y,v) are also solutions of (4). Once properly normalized, since ψ∗(−v) = ψ∗(v), we deduce that ψη(y,v) = ψη(y,−v) = ψη(−y,v). Let us collect our main Assumptions: (H1) The velocity variables v lies in a compact, symmetric set V of Rd equiped with a symmetric probability measure ν. There exist constants C, γ > 0 such that ν({v ∈ V, |v · ξ| ≤ h}) ≤ Chγ, for all ξ ∈ Sd−1, h > 0 . In particular, ν({v ∈ V, |v·ξ| = 0}) = 0 for all ξ 6= 0 so that v ·ξ = 0 a.e. in v implies ξ = 0 . (H2) The source term S ∈ H4(Rd(C∞(Y),L2(V))) (H3) The scattering coefficient σ is in C0(Vv ×Vv′;Cp∞er(Rdy)). It is bounded from above and below by positive constants and is 1-periodic with respect to the variable y. The coefficient Σ(y,v) is defined implicitly in (7). (H4) The symmetry relations (9) hold so that the uniquely defined (after proper nor- malization) solutions ψ∗(v), ψ(y,v) and ψη(y,v) of (10), (7), and (4), respec- tively, satisfy ψ∗(−v) = ψ∗(v) and ψ(y,v) = ψ(−y,v) = ψ(y,−v) as well as ψη(y,v) = ψη(−y,v) = ψη(y,−v). We want to stress again that the limit η → 0 is singular. The limit of ψη as η → 0 is therefore not necessarily equal to ψ(y,v). We are now ready to state our main results on the Hilbert expansion of the solution fε,η := fη of (1). The main result of this paper α is the following theorem, which provides a strong convergence result for the corrector to homogenization theory: Theorem 2.1 Let fε,η be the solution of (1). Then the following expansion holds fε,η − n0,0(x)ρ0(ηx)ψ(ηx,v) ε ε (cid:13) (cid:13) (cid:13) − η n0,0(x)Q−1 −v·∇ (ρ0(ηx)ψ(ηx,v)) +n0,1(x)ρ0(ηx)ψ(ηx,v) y ε ε ε ε (13) h i (cid:0) (cid:1) ε − θ−1(y)ψ(ηx,v)·∇ n0,0(x)+n1,−1(x)ρ0(ηx)ψ(ηx,v) = o(η + ε) ηh ε x ε ε i(cid:13)L2(Rd×V) η (cid:13) where ψ(y,v) and ψ∗(v) are solutions of (10) and (7), respectively(cid:13), and Q−1 is defined in (11). The microscopic density ρ0(y) is the unique solution of the elliptic equation L(ρ0) = 0 with the normalization ρ0(y)dy = 1, Z Y where L(ρ) = − ψ∗(v)v·∇ (Q−1(v ·∇ (ψ(y,v)ρ(y))))dν(v). y y Z V The function θ−1 is given by θ−1 = L−1 (vQ−1(−v ·∇ ρ0(y)ψ(y,v))−v ·∇ (Q−1(vψ(y,v))ρ0(y)))ψ∗(v)dν(v) , y y (cid:18)Z (cid:19) V 5 with L−1 defined in Proposition A.1 below on functions in L2(Y) that average to 0 on Y and the macroscopic density is given by the diffusion equation n0,0(x)−∇ ·(D·∇ n0,0(x)) = S(x,y,v)ψ∗(v)dydν(v). x x Z Z Y V The diffusion coefficient in the preceding equation is given by the expression D = (χ∗0(y,v)⊗vρ0(y)ψ(y,v)+θ∗−1(y)ψ∗(v)⊗vQ−1(−v∇ (ρ0(y)ψ(y,v)))dν(v)dy y Z Z V Y in which we have defined χ∗0 = Q∗−1 vψ∗ +v ·∇ (θ∗−1ψ∗) y (cid:0) (cid:1) θ∗−1 = L∗−1 ψ(y,v)v·∇ Q∗−1 vψ∗(v) dν(v) . y (cid:16)ZV (cid:0) (cid:1) (cid:17) where L∗−1 is also defined in Proposition A.1. The correctors n0,1(x) and n1,−1(x) satisfy the same elliptic equation as n0,0(x) with different source terms. Their expression is given explicitly in Proposition 6.2 below. Before proving this theorem, we make a few remarks. Remark 2.2 The leading term in the expansion of fε is given by n0,0(x)ρ0(x)ψ(x,v). α α α The behavior at the macroscopic level is given by n0,0(x), the solution of a standard diffusion equation. The microscopic level is given by the product of two terms. The first contribution to the product is the standard local solution ψ(y,v), which indicates how particles are distributed in the v variable for each y ∈ Y. The second, less standard, contribution is given by ρ0(x) and indicates how the local (for each y) equilibria are α related to one-another to generate a global equilibrium at the level of the cell Y. Remark 2.3 The above expansion implies that when η ≪ ε, then thecorrector isgiven η by ηx ηx ηx θ−1ψ( ,v)·∇ n0,0(x)+n1,−1(x)ρ0( )ψ( ,v). x ε ε ε This is a regime of (relatively) low scattering where the corrector to homogenization (characterized by a term linear in ∇n0,0) dominates. The contribution n1,−1 provides a correction to the influence of the local equilibria at each y ∈ Y to a global equilibrium on Y. When ε ≪ η, the corrector is given instead by η ηx ηx ηx ηx n0,0(x)Q−1 −v ·∇ (ρ0( )ψ( ,v)) +n0,1(x)ρ0( )ψ( ,v). y ε ε ε ε (cid:0) (cid:1) This is the regime of high scattering, where the correction to approximating the trans- port solution by a diffusion approximation dominates the correction coming from the homogenization procedure. The passage from local to global equilibria on Y generates a corrector described by n0,1(x). The rest of the paper is devoted to the proof of the theorem. 6 3 A priori estimates We start with an estimate that controls the remainder terms in the Hilbert expansion: Proposition 3.1 Let fε be the solution of (1). Then we have: α 1 fε kfεk+ kfε −ψη αk ≤ C kεS k+kS ψη∗k , (14) α ε α αψη α α α α (cid:16) (cid:17) where k·k is the L2(Rd ×V)-norm and for u ∈ L2(Rd ×V) we have defined 1 u¯(x) = u(x,v)dν(v). (15) |V| Z V Proof. We verify that (ηv·∇ +Qη)1 = 0, (−ηv ·∇ +Qη∗)(ψηψη∗) = 0, Y ×V, (16) y y where we have defined the rescaled transport operator σ(y,v′,v)ψη(y,v′) Qηu(y,v) = u(y,v)−u(y,v′) dν(v′). (17) Z ψη(y,v) V (cid:2) (cid:3) The reason for introducing the operator Qη is that 1 x (v·∇ + Qη +ε)u = εF (x,v) := εF(x, ,v), (18) x ε α α α where we have defined fε(x,v) S(x,y,v) u(x,v) = α , F(x,y,v) = , (19) ψη(x,v) ψη(y,v) α and σ(x,v′,v)ψη(x,v′) Qηu = α α u(x,v)−u(x,v′) dν(v′). (20) α Z ψη(x,v) V α (cid:2) (cid:3) Define the operator 1 Tη = v ·∇ + Qη. (21) α x ε α Then we recast (18) as (ε+Tη)u = εF . We verify that α α Qη(h) := (Tηh,ψηψη∗h) α α α α 1 x |h(x,v)−h(x,v′)|2 (22) = σ( ,v′,v)ψη(x,v′)ψη∗(x,v) dxdν(v)dν(v)′. ε Z α α α 2 This comes from the fact that v ·∇ (ψηψη∗) = 1Qη∗(ψηψη∗) so that x α α ε α α α 1 h2 Qη(h) = (Qη(h)h−Qη ,ψηψη∗). α ε α α 2 α α 7 Let ρ = ρ(x) and s = u − ρ for an arbitrary ρ(x) independent of v. Then we find that Qη(ρ) = 0 and more importantly that Qη(u) = Qη(s). Now for s such that α α α sdν(v) = 0, we find that V R β Qη(s) ≥ ksk2, (23) α ε for some β > 0 as is clear from (22) provided that ψηψη∗ is bounded from below by α α a positive constant uniformly in η. We thus define ρ = u¯ and s = u − u¯ so that u¯ is independent of v and s mean zero in v. Multiplying (18) by uψηψη∗ and integrating α α yields ε(u,uψηψη∗)+Qη(s) = (εF ,uψηψη∗). (24) α α α α α α As a consequence, 1 εkuk2+ ksk2 ≤ |(εF ,sψηψη∗)|+|(εF ψηψη∗,ρ)|. (25) ε α α α α α α From this, we deduce the a priori estimate 1 kuk+ ku−u¯k ≤ C kεF k+kF ψηψη∗k . (26) α α α α ε (cid:16) (cid:17) In the variables fε, this is equivalent to (14). α 4 Expansion in ε To emphasize the dependency in η, let us denote fε,η := fε the solution of (1). As in α the standard derivation of diffusion approximations, we first expand fε,η in powers of ε at a fixed (arbitrary) value of η. We prove the following proposition. Proposition 4.1 The solution fε,η can be expanded as follows ηx ηx ηx ηx fε,η = n0,η(x)ψη( ,v)+εf1,η(x, ,v)+ε2f2,η(x, ,v)+ε3f3,η(x, ,v)+rε,η(x,v) ε ε ε ε (27) where we have defined: f1,η = Tη−1(−v ·∇ (n0,ηψη))+n1,ηψη x f2,η = Tη−1(−v ·∇ n1,ηψη)+f2,η x f2,η = Tη−1(S −v ·∇ Tη−1(n0,ηψη)−f0,η) x f3,η = Tη−1(−v ·∇ f2,η −f1,η). x The operator Tη−1 is defined in (12). The density n0,η satisfies the diffusion equation n0,η −∇ · vχη(y,v)ψ∗(v)dν(v)dy∇ n0,η = S(x,y,v)ψη∗(v)dν(v)dy (28) x x Z Z Y×V Y×V where χη = Tη−1(vψη) and the density for the correcting term n1,η satisfies n1,η−∇ · vχη(y,v)ψ∗(v)dν(v)dy∇ n1,η = − v·∇ f2,η(x,y,v)ψ∗(v)dν(v)dy. (29) x x x Z Z Y×V Y×V 8 The remainder term rε,η satisfies the following estimate: ||rε,η||L2 ≤ ε2||v·∇xf3,ηψη∗||L2L∞L2 +ε3||v·∇xf3,η||L2L∞L2 xv x y v x y v (30) +ε2||f2,η||L2L∞L2 +ε3||f3,η||L2L∞L2. x y v x y v Proof. We propose the following ansatz for fε,η : ηx ηx ηx ηx fε,η = f0,η(x, ,v)+εf1,η(x, ,v)+ε2f2,η(x, ,v)+ε3f3,η(x, ,v)+rε,η(x,v). ε ε ε ε We plug the ansatz into the transport equation and equate like powers of ε to obtain the following sequence of equations. At the leading order, we obtain Tηf0,η = 0 which yields that f0,η = n0,η(x)ψη(y,v). We recall that Tη = ηv·∇ +Q. At the next order, f1,η satisfies y Tηf1,η = −v ·∇ f0,η(x,y,v) = −ψη(y,v)v·∇ n0,η(x). x x We can then rewrite f1,η as f1,η(x,y,v) = −χη(y,v)·∇ n0,η(x)+n1,η(x)ψη(y,v) x by defining χη = Tη,−1(vψη), the unique solution to Tηχη = vψη such that χη(y,v)ψ∗(v)dydν(v) = 0. Z Z V Y We can apply the inverse operator Tη,−1 defined in (12) to the source term vψη because the following standard no-drift condition is satisfied: vψη(y,v)ψ∗(v)dν(v)dy = 0, Z Z V Y thanks to assumption (H4). The second-order equation reads Tηf2,η = −f0,η(x,y,v)+S(x,y,v)−v ·∇ f1,η(x,y,v) x and the compatibility equation that the right-hand side must satisfy to be in the domain of definition of Tη,−1 gives the diffusion equation (28) for n0,η. 2,η Introduce f the solution to Tηf2,η = −f0,η(x,y,v)+S(x,y,v)−v ·∇ (−χη(y,v)·∇ n0,η(x)). x x The third-order equation is the following Tηf3,η = −v ·∇ f2,η(x,y,v)−f1,η(x,y,v). x and the corresponding compatibility equation gives a diffusion equation for n1,η n1,η(x)−∇ · vχη(y,v)ψ∗(v)dν(v)dy∇ n1,η(x) = − v·∇ f2,η(x,y,v)ψ∗(v)dν(v)dy x x x Z Z Y×V Y×V 9 since ψ∗(v)χη(y,v)·∇ n0,η(x) = 0. x Z Z Y V With the above expressions, we verify that remainder term now satisfies that: 1 1 ηx ηx ηx rε,η+ v·∇ rε,η+ Q(rε,η) = −ε2f2,η(x, ,v)−ε3f3,η(x, ,v)−ε2v·∇ f3,η(x, ,v). x x ε ε2 ε ε ε Thanks to the estimate (14) proved in the previous section, we obtain (30). In order to analyze the behavior of the terms f0,η, f1,η, f2,η and f3,η as η → 0, we need to study some auxiliary equations. Such studies are conducted in the following section. 5 Expansion of the auxiliary functions This section is devoted to the asymptotic expansion as η → 0 of the three cell functions ψη, χη, and χη∗. 5.1 Expansion of ψη Recall that ψη satisfies ηv ·∇ ψη +Q(ψη) = 0 and ψη(y,v)ψ∗(v)dν(v)dy = 1. y Z Z Y V We prove the following expansion Proposition 5.1 The function ψη, solution to (4) satisfies ψη(y,v) = ψ0(y,v)+ηψ1(y,v)+η2ψ2(y,v)+r˜η(y,v), ||r˜η|| ≤ Cη3, (31) p p Hk(Y,L2(V)) with ψ0(y,v) = ρ0(y)ψ(y,v), ψ1(y,v) = Q−1(−v ·∇ ψ0(y,v)) (32) y ψ2(y,v) = Q−1(−v ·∇ ψ1(y,v))+ρ2(y)ψ(y), y where L(ρ0) = 0 with ρ0(y)dy = 1 (33) Z ρ2 = −L−1 ψ∗(v)v ·∇ (Q−1(−v ·∇ (Q−1(−v ·∇ ψ1(y,v)))))dν(v) . (34) y y y (cid:18)Z (cid:19) V The operator L and its inverse defined on functions in L2(Y) with vanishing average over Y are defined in Proposition A.1 in the appendix. From this expansion, we deduce that: ||ψη||L2L∞ ≤ C and || vψηψ∗dν(v)||L∞ ≤ Cη. v y Z y V Moreover, we find that ψ0 and ψ2 are even functions and ψ1 is an odd function in the variable v. 10